A topological degree counting for some Liouville systems of mean field type
Journal
Communications on Pure and Applied Mathematics
Journal Volume
64
Journal Issue
4
Pages
556-590
Date Issued
2011
Author(s)
Zhang, L.
Abstract
Let A = (aij)n×n be an invertible matrix and A-1 = (aij)n×n be the inverse of A. In this paper, we consider the generalized Liouville system 0.1 Δgui+∑j=1n aijρj(hj eu-j/∫hjeu-j-1) = 0 in M, where 0 < hj ∈ C1(M) and ρj ∈ R{double-struck}+ and prove that, under the assumptions of (H1) and (H2) (see Introduction), the Leray-Schauder degree of (0.1) is equal to (-Χ(M)+1)\...(-Χ(M)+N)/N! if ρ = (ρ1,..., ρn) satisfies 8ΠNsumi=1nρi< ∑1≤i, j≤n aij ρiρj<8π(N+1) ∑i=1nρi.Equation (0.1) is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of the nonlinear function Φρ: Φρ(u) = 1/2 ∫M∑1≤i,j≤ n aij∇g ui · ∇g uj+∑i=1n∫M ρi ui - ∑i=1n ρi log ∫M h-i e u-i. The Liouville system (0.1) has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems. © 2010 Wiley Periodicals, Inc.
Type
journal article